Quantum estimation theory, as the mathematical language of quantum metrology [1]}, [2]}, [3]}, is of critical importance in the development of high-precision devices in several technological areas. Its main objective is to realize high precision measurements by estimating the unknown parameters that specify a given quantum system using quantum effects. In this sense, it aims at developing new methods to improve the accuracy limit of physical parameters beyond classical metrological methods [4]}, [5]}, [6]}, [7]}. In short, it is interested in achieving the highest possible accuracy in various parameter estimation tasks, and in finding measurement schemes that achieve this accuracy. Originally, metrology focused on measurements made using classical systems, such as mechanical systems described by classical physics or optical systems modeled by classical wave optics [8]}. We must therefore ask how can we deduce the limits of accuracy of parameter estimation, as well as methods that can improve accuracy in quantum systems ? Is there a fundamental limit ? In the classical scheme, the first answers appeared around the 1940s with the seminal paper of Rao [9]} and Cramer [10]}, who independently found a lower bound on the variance of an arbitrary estimator. These results were extended to the multiparametric strategy by Darmois [11]}. This limit, generally called the Cramér-Rao bound, is closely linked to the Fisher information, introduced by Fisher in the 1920s [12]}. The Fisher information thus plays a central role in estimation theory. Its maximization on all possible quantum measures defines the quantum Fisher information and provides a quantum lower bound to the Cramér-Rao bound [13]}, [14]}.
